Until now we have been studying mostly quantum theories and physical phenomena
which were discovered relatively a long time ago.
So I think maybe the latest theoretical development we talked about, the path
integral itself goes back to the 40's, 1940's.
Today in this video, towards the end of the video I'm going to tell you a, about a
more recent theoretical development. A so-called theory of quantum localization
or more precisely weak localization. And that will do so using the main ideas
of the Feyman path integral. So I should mention that the 4 theory of
quantum localization is an extremely complicated mathematical theory and it
could be a subject though for separate course.
So I don't have time for that and obviously, we don't have the mathematical
sort of background to talk about it. But it turns out that some basic ideas
behind this phenomenon can be understood using the basic concepts behind the
Feynman path integral, which comes handy here.
So just in a few words, what I'm going to be taking about here is the following.
So we know that in solid state physics, materials sort of can be divided into
roughly two categories. Insulators and metals.
They're also semiconductors but from this perspective, they're much like insulators.
So and if you, what insulators are, if you apply a weak electric field the electrons
which exist inside the material don't move and so there is no current really in
response to electric field. So, If you think about a piece of food or
something like that. So we obviously understand it's an
insulator, it's not going to be a very useful electric component.
On the other hand, if you have a metallic system, let's say a piece of copper, and
if we apply electric field or voltage there will be current in response to this
electric field. And so this characterizes a metal, so in
[unknown] there is some intersections disorder, impurities , dislocation all
kind of things and these imperfections. They lead to a finite resistance.
Okay, and so, the next slide will tell you about how this resistance comes about.
But later on we will see how essentially the classical theory of transport that
applies to most metal at high temperature get more defined as you go to lower and
low temperatures quantum effects come into play, the way of nature for instance, the
play, and this is very interesting and it turns out that there under certain
circumstances you can get complete completely different behavior at low
temperatures. Metal actually would become insulators and
this process of trapping electrons due to quantum mechanical effects is called
mobile localization. Of course, it's a very mathematical
theory, as I told you, but we will try to get some idea about where it comes from.
But before discussing a complicated quantum theory, let me first talk about a
basic classical, very simple, theory of how we can understand electrical transport
in a metal. So here this black dot labels an electron
which can move around in, in a piece of metal.
And these red dots here, large dots, they represent some impurities, both of which
the electron can scatter. And that's exactly what it does, as it
moves around so it gets scattered by these random impurities and soon the trajectory
of this electron is this diffused trajectory, so it's random walk.
Of course it's not just one, there are many, many electrons in the metal and they
all experience this random walk. Which in the absence of an electric field
just averages out to zero. There is no net current, there is no
preferential direction, there is nothing which breaks the symmetry.
But now, if we let's say apply electric field in this direction the electrons
which have a negative charge are going to move in the opposite direction
preferentially. They are still going to move around in all
directions, but there will be sort of a drift in one particular direction, and
this will give rise to an a current. Now it turns out that the first theory of
this conductivity of this current in a metal was put together back in 1900 and
this [unknown] theory is the so called Drude theory which is a very simplistic
theory that I am going to explain now. So, basically the Drude model assume that
we can just view the [unknown] as a classical particle which there [inaudible]
was the classical equations of motion, the Newton equation.
And in the presence of electric field so that the charge, let's see if the charge
is q, so it experiences this force due to the electric field, and these impurities
all these imperfections, he just replaced with the friction force.
And this friction force was assumed to be proportional to the velocity.
Lets say, it sort of makes sense if you have a viscous fluid for instance, and if
you put. An object is going to move in this fluid
so its going to experience some force, a friction which is proportional eh
proportional to the velocity and with the minus sign of course and there is some
coefficient. So velocity is momentum divided by the
mass. So we can write instead of writing the
friction force this way we can write it as minus the p over t of some tau which is
the time scale which describes. When we have the [unknown] in this prose.
So in some sense you may think about the stall as a typical time between collision
for the electron. But in the drude model it was just a sort
of in the logical parameter of the described friction, or momentum relaxation
of this electron. And now what he did, he simply wrote down
this model, and so of course we can write the left-hand side of the Newton equation
as dp/dt, and so the right-hand side just putting this together is going to be the
electric force and this friction force. And in equilibrium, we demand that there
is no acceleration of electrons. So if we, sort of, if we, if we take let's
say, a piece of metal, apply some electric field, eventually it's going to come to
some sort of state of equilibrium. There's going to be a current, but there's
not going to be indefinite acceleration of electrons.
So they're going to come to a certain sort of steady state, and this steady state
will determine the current. Now and from here what we can do we can
just solve for, for the velocity so momentum is equal from this equation is
going to be equal q times E times tal and velocity therefore is going to be just
momentum divided by the map. And at this stage what we can do, so we
can write the current. So, the current is simply the charge of,
of the carriers in this case electrons, times the density of these carriers how
many of those participate in transport and v is the velocity, which we have just
determined. And so, if we put everything together,
we're going to get this expression. So, it's going to be this [unknown] of
friction, which is called conductivity and times the electric field.
So, by definition actually the conductivity is the coefficient of
proportionality between the current and the electric field.
So this expression, this nq squared tau over m is the so-called drude
conductivity, which as you can see is a purely classical entities.
So there is no quantum mechanics here whatsoever.
Now, embarrassing thing that I have to admit at this stage, is that almost all
sort of useful theories of metals, of transporting metals rely on this drude
equation up to this date. So since, for more than 100 years we've
been using this equation and in most cases actually it works fine.
But fortunately well for us scientists otherwise it would have been really
boring. So if you go to lower temperatures, the
oxidation becomes a little more tricky. And this is because quantum mechanics
become, becomes important. So quantum mechanics come into play.
And what it means is that, in this picture, for instance, what, what, how we
can envision the appearance of quantum effects is that instead of just straight
trajectories of classical particles, we're going to have waves.
Bouncing ground of, of this incurities and so this waves are going to interfere with
one at, with each other in some sense. So its going to be the particles going to
interfere with itself and this quantum interference [inaudible] are going to,
going to modify this picture a very significantly.
So now let us look at the mo, electron motion in methalanic disorder, a metal
from a slightly different perspective, actually from a completely different
perspective because now we want to exclusively to chernon quantum mechanics.
And we will use quantum mechanics [unknown] of quantum mechanics due to
Feynman and ask the question of what is the probability of for an electron to
diffuse from an initial point into the final point F for this forest of impurity?
And according to the Feynman this probability can be written as the absolute
value squared of a sum of quantum mechanical amplitudes associated with all
possible classical paths. So this diffusion trajectory here, I'm
just showing two such possible trajectories.
You know this blue one and the black one and this guys are going to interfere with
one another and this interference suppose instead if I call this trajectory,
trajectory 1, and this is going to be trajectory 2 so, the interference term
which is which is going to appear due to this from this expression is going to
involve a product of this black exponential [unknown] h bar times the Lew
exponential which is complex conjugate [unknown] divided h bar plus its complex
conjugate. So, oh no, the interference term coming
from these two particular trajectories, again this is just two examples.
There are infinitely many such trajectories and infinitely many such
term. But in any case, each of these terms is
going to produce this sort of cosine of s1 minus s2 divided by h bar.
So this is basically the quantum interference terms.
And the terms which don't involve the cross product of different trajectories I
classical terms. So in some sense, these are the terms.
Which are responsible for the judic conductivity, the classical judic
conductivity that we obtained in the previous discussion, okay and so now the
question of what want what is, what is quantum mechanics doing, what are the
quantum directions goes down to figuring out what this terms are.
Now in the beginning of this video I mentioned something about temperature,
said well at high temperature so this terms don't really matter at low
temperatures. They start to measure and the reason this
is the case because at high temperatures what the temperature really is, it
involves lot of motion which occurs around these electrons which are sort of doing
their thing moving around, diffusing around.
But there is also let's say lattice vibrations, all kinds of things and this
sort of shaking of the electron disturbs the phases, so in some sense this is what
is called sometimes the quantum mechanical literature, this is called the phasing.
So basically here in this expression whatever theory we're going to be doing
we're going to determine these actions We're going to assume that phase one of
mechanical phase is going to remain constant during the , or unperturbed more
to say during the motion of election or along these trajectories, but it turns out
that even if we assume that there is no disturbance of the phases and no new
phasing. So classical physics in some sense
protects itself in that this quantum interference theorems tends to cancel each
other out and completely disappear from the picture.
And we'll actually have to search really hard in order to find the [inaudible]
where they start being essential. In order to see all this let me estimate
the typical action that appears in this equation.
So remember that. Well, at this stage, we're just talking
about particles bouncing, bouncing around from these impurities.
But otherwise, in between each scattering they are free particles.
So their energy is essentially just mv squared over 2.
The kinetic energy of a free particle. And therefore, in order to estimate the
action. We can just say that well of the, by, by
the at the projector. So now using this estimate we can also
estimate the typical a quantum interference terms which are which I
described by the sum of over all pairs of non-equivalent different trajectories L1
and L2 and now instead of the action S we write exclusively at the typical momentum
which by the way is called Feyman momentum divided by.
H bar times the difference in length between the two trajectories under
consideration. Or we can write the [inaudible] to be as
the cosine sum over l one not equal to l two cosine of typical momentum u times
delta l, divided by h bar. So it turns out that at these argumental,
the cosine in the typical metal is actually very large and this is because
the typical velocity of electron in a metal is less than the percent of the
speed of light which I can't[INAUDIBLE] the speed of light which still is pretty
fast and therefore the typical firmly to the typical wavelength turns out to be
less than a. [unknown] minus nine meters, which is
usually much smaller than the distance between these red dots here.
So I'm just giving you facts, it's not by any means obvious, it doesn't follow from
the previous discussion and I'm just giving you certain information and the
bottom line here is that the typical wave length, in some sense you might think
about action's really being a wave, and this wavelength is much, much smaller than
the distance between these red dots. And so the interference terms are not very
important. How it plays out here is that?
So, this typical [inaudible] of delta l divided by essentially, this lambda is
very large. And so, you have different trajectories
contributing to this sum. And so you have cosines of different
phases, completely random phases. And cosine as we know is the function
which can be a positive or it can be negative depending so if this our cosine
of pi, this is pi. So it can be positive or negative.
So if we completely pick this phase is completely in random.
So the average value of the cosine is going to be 0.
So the terms with different signs of cosine basically cancel each other out and
this is exactly. What happens with quantum interference
terms? And so, seemingly classical physics
survives, and we can just completely forget about this quantum interference
phenom. So fortunately, it's not quite the case.
It's almost the case, actually in [inaudible] disorder [inaudible] you see
there is a leap year when an electron can actually travel in the clockwise
direction, and this is going to be our path number one.
Or it can decide to travel in the opposite direction, counterclockwise, in which case
it's going to be a path number two. And the two paths, they're going to be
different paths but they're going to have exactly the same way.
And therefore the quantum interference terms responding to this pass are going to
survive the this phase difference will be 0 and this terms are going to play
[inaudible] significantly and so this is where we going to task and this second
part of this of this video. Now going back to little bit to physics in
order just to get give you an intuition of what, what, what it has to do with
conductivity. So we can sort of can individually
understand that the easier it is for an electron to move from an initial point to
some just and final point battery, this material conducts.
Okay the battery diffuses the battery it conducts and the classical level it is
just [inaudible] by this [inaudible]. On the other hand the easier for the
electron to return to the same point the, the more often it just sort of goes back
and back. So the worst conductor it is.
So in some sense the probability to have this self crossing trajectories, should in
general, by general arguments reduce the conductivity.
And this is exactly what happens, and we'll most cases.
And therefore, it is called [inaudible]. Basically tries to bring electron back to
where it came from.